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On the characteristic polynomial of a supertropical adjoint matrix
Linear Algebra and its Applications 499 (2016) 26–30
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
On the characteristic polynomial of a supertropical adjoint matrix Yaroslav Shitov National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia
a r t i c l e
i n f o
Article history: Received 30 January 2015 Accepted 3 March 2016 Available online 9 March 2016 Submitted by R. Brualdi MSC: 15A80 Keywords: Tropical algebra Matrix theory
a b s t r a c t Let χ(A) denote the characteristic polynomial of a matrix A over a field; a standard result of linear algebra states that χ(A−1 ) is the reciprocal polynomial of χ(A). More formally, the condition χn (A)χk (A−1 ) = χn−k (A) holds for any invertible n × n matrix A over a field, where χi (A) denotes the coefficient of λn−i in the characteristic polynomial det(λI − A). We confirm a recent conjecture of Niv by proving the tropical analogue of this result. © 2016 Elsevier Inc. All rights reserved.
1. Preliminaries The supertropical semifield is a relatively new concept arisen as a tool for studying problems of tropical mathematics [4]. The supertropical theory is now a developed branch of algebra, and we refer the reader to [5] for a survey of basics and applications. Our arguments make use of some other structures including fields and polynomial rings over them, so it will be convenient for us to work with slightly unusual equivalent description of the supertropical semifield. In particular, we will denote the tropical operations by ⊕ and to avoid confusion with standard operations + and · over a field. For the same E-mail address: [email protected]. http://dx.doi.org/10.1016/j.laa.2016.03.005 0024-3795/© 2016 Elsevier Inc. All rights reserved.
Y. Shitov / Linear Algebra and its Applications 499 (2016) 26–30
27
reason, we will use the notation ui in supertropical setting while ui will denote the power of an element of a field. Similarly, we will denote the supertropical determinant by det◦ , reserving the notation det for usual determinant over a field. Let us recall the definitions of concepts mentioned above. Let (G, ∗, 1, ≤) be an ordered Abelian group (in multiplicative notation), and G (0) and G (1) be two copies of G. We consider the semiring S = G (0) ∪ G (1) ∪ {0} with two commutative operations, denoted by ⊕ and . Assume i, j ∈ {0, 1}, s ∈ S, a, b ∈ G, and let a < b; the operations are defined by 0 ⊕ s = s ⊕ 0 = s, 0 s = s 0 = 0, b(j) ⊕ a(i) = a(i) ⊕ b(j) = b(j) , b(i) ⊕ b(j) = b(0) , a(i) b(j) = (a ∗ b)(ij) . One can note that S is isomorphic to the supertropical semifield, and the elements from G (0) and G (1) correspond to the ghost and tangible elements, respectively. In particular, the elements 0 and 1(1) are neutral with respect to ⊕ and , respectively. We recall that the mapping ν sends a(i) to a ∈ G and 0 to 0; the elements c, d ∈ S are ν-equivalent whenever ν(c) = ν(d), and we write c ≈ν d in this case. Also, we write c |= d if either c = d or c = d ⊕ g, for some ghost element g; this relation is known as the ghost surpassing relation, and it is one of fundamental concepts that replaces equality in many theorems taken from classical algebra [5]. By ui we denote the ith supertropical power of u, that is, the result of multiplying u by itself i times. The operations on S are now defined, and we can speak of vectors, matrices and polynomials over S; we refer the reader to [2] for a thorough discussion of the matrix and polynomial algebras over a semiring. We define the operations with supertropical polynomials and matrices in the same way as over fields but with standard arithmetic operations replaced by their supertropical counterparts ⊕ and . In particular, the supertropical identity matrix I◦ has elements 1(1) on the diagonal and 0’s everywhere else. 2. The result Let A = (aij ) be a supertropical matrix; its determinant is det◦ A =
a1σ(1) . . . anσ(n) ,
σ∈Sn
where Sn denotes the symmetric group on {1, . . . , n}. The matrix A is said to be nonsingular if det◦ A is tangible; equivalently, A is non-singular if det◦ A has a multiplicative inverse in S. The (i, j)th cofactor of A is the supertropical determinant of the matrix obtained from A by removing the ith row and jth column. By adj◦ A we denote the adjoint of A, that is, the n × n matrix whose (i, j)th entry equals the (j, i)th cofactor. By χk◦ (A) we denote the coefficient of λ(n−k) in the characteristic polynomial det◦ (A ⊕ λ I◦ ); for k = 0, this quantity equals the supertropical sum of all principal k × k minors of A. The following has been an open problem until now. Conjecture 1. (See [1, Conjecture 6.2].) Let A ∈ S n×n be a non-singular matrix. Then, (k−1) χk◦ (adj◦ A) |= (det◦ A) χn−k (A) holds for all k ∈ {0, . . . , n}. ◦
Y. Shitov / Linear Algebra and its Applications 499 (2016) 26–30
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Actually, Niv formulates this conjecture in a slightly different but equivalent way. If A is non-singular, then its pseudoinverse is defined as A∇ = (det◦ A)(−1) (adj◦ A). (The motivation behind the definition of A∇ and a discussion of its properties can be found in [3].) Multiplying both sides of equality in Conjecture 1 by (det A)(1−k) , one gets (det◦ A) χk◦ (A∇ ) |= χn−k (A), which is exactly the formulation given by Niv. ◦ 3. Our strategy We note that k = 0 is a trivial case in Conjecture 1, and in what follows we assume that k ∈ {1, . . . , n}. We consider the n × n matrix V whose (i, j)th entry is a variable (vij ), and we define the polynomials αkn , βkn ∈ S[v11 , v12 , . . . , vnn ] as αkn = χk◦ (adj◦ V ),
(k−1)
βkn = (det◦ V )
χn−k (V ). ◦
Let us now fix n and k; slightly abusing the notation, we write simply α, β instead of αkn , βkn . Note that any coefficient of α and β is either 1(0) or 1(1) ; we define γ ∈ S[v11 , . . . , vnn ] as the supertropical sum of those monomials that appear in β with tangible coefficients. Remark 2. Let f ∈ S[v11 , . . . , vnn ] be a supertropical polynomial and M ∈ S n×n a supertropical matrix. By f (M ) ∈ S we denote the value of f computed at M ; in other words, f (M ) equals the value of expression obtained from f by replacing every variable vij by the (i, j)th entry of M . Our proof goes as follows. First, we prove that the ghost coefficients of β are inessential. That is, we show in Claim 3 that β(A) = γ(A) if A ∈ S n×n is a non-singular matrix. Further, we employ the standard result mentioned in the abstract. We prove in Claim 4 that any monomial appearing in either α or β with the tangible coefficient appears in both of them. Combining these facts in Claims 5 and 6, we deduce that the equality α = γ ⊕ ρ is true for some polynomial ρ, and β(A) = α(A) ⊕ s holds for some s ∈ S whenever α(A) is tangible. Finally, these equalities allow us to show that α(A) = β(A) ⊕ ρ(A) and that ρ(A) is either tangible or does not exceed β(A), which proves Conjecture 1. 4. The proof Claim 3. If A ∈ S n×n is a non-singular matrix, then β(A) = γ(A). Proof. By definition, β is the supertropical sum of monomials mμ = m1μ m2μ , where m1μ
=
n i=1
viσ1 (i)
...
n i=1
viσk−1 (i)
, m2μ =
j∈J
vjτ (j) ,
Y. Shitov / Linear Algebra and its Applications 499 (2016) 26–30
29
over all families μ = (σ1 , . . . , σk−1 , J, τ ) such that σ1 , . . . , σk−1 ∈ Sn , a subset J ⊂ {1, . . . , n} has n − k elements, and τ is a permutation of J. (In particular, we have m1μ = 1(1) if k = 1 and m2μ = 1(1) if k = n.) Assume β(A) = 0. Suppose that there exist distinct families μ and μ = (σ1 , . . . , , J , τ ) satisfying mμ = mμ and β(A) ≈ν mμ (A). Then, since there is a unique σk−1 n permutation σ such that i=1 aiσ(i) ≈ν det◦ A, we have that all σt and all σt are equal to σ. In particular, the monomials m1μ and m1μ are equal, which implies m2μ = m2μ . Note that the latter condition in turn implies J = J and τ = τ , a contradiction. Therefore, either β(A) = 0 or ν(m(A)) = ν(β(A)) holds for any monomial m that appears in β with ghost coefficient. This means that we can remove all these monomials from β without changing the value of β(A). 2 k11 k12 knn v12 . . . vnn appears with a tangible coefficient in Claim 4. If a monomial v11 either α or β, then it appears in both α and β with coefficients different from 0.
Proof. Let X = (xij ) be a matrix whose entries are variables of the polynomial ring k−1 n−k C[x11 , . . . , xnn ], and define ϕ = χk (adj X), ψ = (det X) χ (X). Let us get rid of brackets by distributivity in standard expressions of ϕ and ψ and denote the expressions we obtain before canceling terms by ϕ0 and ψ0 , respectively. Replacing every monomial k11 k12 knn nn m = ±xk1111 xk1212 . . . xknn by μm = 1(1) v11 v12 . . . vnn
in ϕ0 and ψ0 , we get α and β. If some monomial μm appears in either α or β with the tangible coefficient, then the corresponding monomial m appears exactly once in either ϕ0 or ψ0 . Since the equality ϕ = (−1)n ψ is true for matrices over a field, m appears in both ϕ0 and ψ0 . In other words, the monomial μm appears in both α and β. 2 Claim 5. Let A ∈ S n×n be a non-singular matrix. If α(A) is a tangible element, then there is s ∈ S such that β(A) = α(A) ⊕ s. k11 knn Proof. By assumption, there is a monomial m = v11 . . . vnn appearing in α with tangible coefficient such that m(A) = α(A). But m appears in β as well by Claim 4, so the result follows. 2
Claim 6. There is a polynomial ρ ∈ S[v11 , . . . , vnn ] such that α = γ ⊕ ρ. Proof. By definition of γ, it is the sum of monomials that appear in β with tangible coefficients. All these monomials appear in α as well by Claim 4. 2 Proof of Conjecture 1. By Claims 3 and 6, there is u ∈ S such that α(A) = β(A) ⊕ u. Claim 5 rules out the case when u is tangible and greater than β(A). It remains to note
30
Y. Shitov / Linear Algebra and its Applications 499 (2016) 26–30
that α(A) and β(A) are, respectively, the left-hand and right-hand sides of the assertion of Conjecture 1. 2 I would like to thank the anonymous reviewers for careful reading of the preliminary versions. Their comments and suggestions lead to a significant improvement in presentation of the paper. References [1] A. Niv, On pseudo-inverses of matrices and their characteristic polynomials in supertropical algebra, Linear Algebra Appl. 471 (2015) 264–290. [2] J.S. Golan, Semirings and Their Applications, Kluwer Acad. Publ., Dordrecht, 1999. [3] Z. Izhakian, Tropical arithmetic and matrix algebra, Comm. Algebra 37 (2009) 1445–1468. [4] Z. Izhakian, L. Rowen, Supertropical algebra, Adv. Math. 225 (2010) 2222–2286. [5] Z. Izhakian, L. Rowen, A guide to supertropical algebra, in: Advances in Ring Theory, in: Trends Math., 2010, pp. 283–302.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
On the characteristic polynomial of a supertropical adjoint matrix Yaroslav Shitov National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia
a r t i c l e
i n f o
Article history: Received 30 January 2015 Accepted 3 March 2016 Available online 9 March 2016 Submitted by R. Brualdi MSC: 15A80 Keywords: Tropical algebra Matrix theory
a b s t r a c t Let χ(A) denote the characteristic polynomial of a matrix A over a field; a standard result of linear algebra states that χ(A−1 ) is the reciprocal polynomial of χ(A). More formally, the condition χn (A)χk (A−1 ) = χn−k (A) holds for any invertible n × n matrix A over a field, where χi (A) denotes the coefficient of λn−i in the characteristic polynomial det(λI − A). We confirm a recent conjecture of Niv by proving the tropical analogue of this result. © 2016 Elsevier Inc. All rights reserved.
1. Preliminaries The supertropical semifield is a relatively new concept arisen as a tool for studying problems of tropical mathematics [4]. The supertropical theory is now a developed branch of algebra, and we refer the reader to [5] for a survey of basics and applications. Our arguments make use of some other structures including fields and polynomial rings over them, so it will be convenient for us to work with slightly unusual equivalent description of the supertropical semifield. In particular, we will denote the tropical operations by ⊕ and to avoid confusion with standard operations + and · over a field. For the same E-mail address: [email protected]. http://dx.doi.org/10.1016/j.laa.2016.03.005 0024-3795/© 2016 Elsevier Inc. All rights reserved.
Y. Shitov / Linear Algebra and its Applications 499 (2016) 26–30
27
reason, we will use the notation ui in supertropical setting while ui will denote the power of an element of a field. Similarly, we will denote the supertropical determinant by det◦ , reserving the notation det for usual determinant over a field. Let us recall the definitions of concepts mentioned above. Let (G, ∗, 1, ≤) be an ordered Abelian group (in multiplicative notation), and G (0) and G (1) be two copies of G. We consider the semiring S = G (0) ∪ G (1) ∪ {0} with two commutative operations, denoted by ⊕ and . Assume i, j ∈ {0, 1}, s ∈ S, a, b ∈ G, and let a < b; the operations are defined by 0 ⊕ s = s ⊕ 0 = s, 0 s = s 0 = 0, b(j) ⊕ a(i) = a(i) ⊕ b(j) = b(j) , b(i) ⊕ b(j) = b(0) , a(i) b(j) = (a ∗ b)(ij) . One can note that S is isomorphic to the supertropical semifield, and the elements from G (0) and G (1) correspond to the ghost and tangible elements, respectively. In particular, the elements 0 and 1(1) are neutral with respect to ⊕ and , respectively. We recall that the mapping ν sends a(i) to a ∈ G and 0 to 0; the elements c, d ∈ S are ν-equivalent whenever ν(c) = ν(d), and we write c ≈ν d in this case. Also, we write c |= d if either c = d or c = d ⊕ g, for some ghost element g; this relation is known as the ghost surpassing relation, and it is one of fundamental concepts that replaces equality in many theorems taken from classical algebra [5]. By ui we denote the ith supertropical power of u, that is, the result of multiplying u by itself i times. The operations on S are now defined, and we can speak of vectors, matrices and polynomials over S; we refer the reader to [2] for a thorough discussion of the matrix and polynomial algebras over a semiring. We define the operations with supertropical polynomials and matrices in the same way as over fields but with standard arithmetic operations replaced by their supertropical counterparts ⊕ and . In particular, the supertropical identity matrix I◦ has elements 1(1) on the diagonal and 0’s everywhere else. 2. The result Let A = (aij ) be a supertropical matrix; its determinant is det◦ A =
a1σ(1) . . . anσ(n) ,
σ∈Sn
where Sn denotes the symmetric group on {1, . . . , n}. The matrix A is said to be nonsingular if det◦ A is tangible; equivalently, A is non-singular if det◦ A has a multiplicative inverse in S. The (i, j)th cofactor of A is the supertropical determinant of the matrix obtained from A by removing the ith row and jth column. By adj◦ A we denote the adjoint of A, that is, the n × n matrix whose (i, j)th entry equals the (j, i)th cofactor. By χk◦ (A) we denote the coefficient of λ(n−k) in the characteristic polynomial det◦ (A ⊕ λ I◦ ); for k = 0, this quantity equals the supertropical sum of all principal k × k minors of A. The following has been an open problem until now. Conjecture 1. (See [1, Conjecture 6.2].) Let A ∈ S n×n be a non-singular matrix. Then, (k−1) χk◦ (adj◦ A) |= (det◦ A) χn−k (A) holds for all k ∈ {0, . . . , n}. ◦
Y. Shitov / Linear Algebra and its Applications 499 (2016) 26–30
28
Actually, Niv formulates this conjecture in a slightly different but equivalent way. If A is non-singular, then its pseudoinverse is defined as A∇ = (det◦ A)(−1) (adj◦ A). (The motivation behind the definition of A∇ and a discussion of its properties can be found in [3].) Multiplying both sides of equality in Conjecture 1 by (det A)(1−k) , one gets (det◦ A) χk◦ (A∇ ) |= χn−k (A), which is exactly the formulation given by Niv. ◦ 3. Our strategy We note that k = 0 is a trivial case in Conjecture 1, and in what follows we assume that k ∈ {1, . . . , n}. We consider the n × n matrix V whose (i, j)th entry is a variable (vij ), and we define the polynomials αkn , βkn ∈ S[v11 , v12 , . . . , vnn ] as αkn = χk◦ (adj◦ V ),
(k−1)
βkn = (det◦ V )
χn−k (V ). ◦
Let us now fix n and k; slightly abusing the notation, we write simply α, β instead of αkn , βkn . Note that any coefficient of α and β is either 1(0) or 1(1) ; we define γ ∈ S[v11 , . . . , vnn ] as the supertropical sum of those monomials that appear in β with tangible coefficients. Remark 2. Let f ∈ S[v11 , . . . , vnn ] be a supertropical polynomial and M ∈ S n×n a supertropical matrix. By f (M ) ∈ S we denote the value of f computed at M ; in other words, f (M ) equals the value of expression obtained from f by replacing every variable vij by the (i, j)th entry of M . Our proof goes as follows. First, we prove that the ghost coefficients of β are inessential. That is, we show in Claim 3 that β(A) = γ(A) if A ∈ S n×n is a non-singular matrix. Further, we employ the standard result mentioned in the abstract. We prove in Claim 4 that any monomial appearing in either α or β with the tangible coefficient appears in both of them. Combining these facts in Claims 5 and 6, we deduce that the equality α = γ ⊕ ρ is true for some polynomial ρ, and β(A) = α(A) ⊕ s holds for some s ∈ S whenever α(A) is tangible. Finally, these equalities allow us to show that α(A) = β(A) ⊕ ρ(A) and that ρ(A) is either tangible or does not exceed β(A), which proves Conjecture 1. 4. The proof Claim 3. If A ∈ S n×n is a non-singular matrix, then β(A) = γ(A). Proof. By definition, β is the supertropical sum of monomials mμ = m1μ m2μ , where m1μ
=
n i=1
viσ1 (i)
...
n i=1
viσk−1 (i)
, m2μ =
j∈J
vjτ (j) ,
Y. Shitov / Linear Algebra and its Applications 499 (2016) 26–30
29
over all families μ = (σ1 , . . . , σk−1 , J, τ ) such that σ1 , . . . , σk−1 ∈ Sn , a subset J ⊂ {1, . . . , n} has n − k elements, and τ is a permutation of J. (In particular, we have m1μ = 1(1) if k = 1 and m2μ = 1(1) if k = n.) Assume β(A) = 0. Suppose that there exist distinct families μ and μ = (σ1 , . . . , , J , τ ) satisfying mμ = mμ and β(A) ≈ν mμ (A). Then, since there is a unique σk−1 n permutation σ such that i=1 aiσ(i) ≈ν det◦ A, we have that all σt and all σt are equal to σ. In particular, the monomials m1μ and m1μ are equal, which implies m2μ = m2μ . Note that the latter condition in turn implies J = J and τ = τ , a contradiction. Therefore, either β(A) = 0 or ν(m(A)) = ν(β(A)) holds for any monomial m that appears in β with ghost coefficient. This means that we can remove all these monomials from β without changing the value of β(A). 2 k11 k12 knn v12 . . . vnn appears with a tangible coefficient in Claim 4. If a monomial v11 either α or β, then it appears in both α and β with coefficients different from 0.
Proof. Let X = (xij ) be a matrix whose entries are variables of the polynomial ring k−1 n−k C[x11 , . . . , xnn ], and define ϕ = χk (adj X), ψ = (det X) χ (X). Let us get rid of brackets by distributivity in standard expressions of ϕ and ψ and denote the expressions we obtain before canceling terms by ϕ0 and ψ0 , respectively. Replacing every monomial k11 k12 knn nn m = ±xk1111 xk1212 . . . xknn by μm = 1(1) v11 v12 . . . vnn
in ϕ0 and ψ0 , we get α and β. If some monomial μm appears in either α or β with the tangible coefficient, then the corresponding monomial m appears exactly once in either ϕ0 or ψ0 . Since the equality ϕ = (−1)n ψ is true for matrices over a field, m appears in both ϕ0 and ψ0 . In other words, the monomial μm appears in both α and β. 2 Claim 5. Let A ∈ S n×n be a non-singular matrix. If α(A) is a tangible element, then there is s ∈ S such that β(A) = α(A) ⊕ s. k11 knn Proof. By assumption, there is a monomial m = v11 . . . vnn appearing in α with tangible coefficient such that m(A) = α(A). But m appears in β as well by Claim 4, so the result follows. 2
Claim 6. There is a polynomial ρ ∈ S[v11 , . . . , vnn ] such that α = γ ⊕ ρ. Proof. By definition of γ, it is the sum of monomials that appear in β with tangible coefficients. All these monomials appear in α as well by Claim 4. 2 Proof of Conjecture 1. By Claims 3 and 6, there is u ∈ S such that α(A) = β(A) ⊕ u. Claim 5 rules out the case when u is tangible and greater than β(A). It remains to note
30
Y. Shitov / Linear Algebra and its Applications 499 (2016) 26–30
that α(A) and β(A) are, respectively, the left-hand and right-hand sides of the assertion of Conjecture 1. 2 I would like to thank the anonymous reviewers for careful reading of the preliminary versions. Their comments and suggestions lead to a significant improvement in presentation of the paper. References [1] A. Niv, On pseudo-inverses of matrices and their characteristic polynomials in supertropical algebra, Linear Algebra Appl. 471 (2015) 264–290. [2] J.S. Golan, Semirings and Their Applications, Kluwer Acad. Publ., Dordrecht, 1999. [3] Z. Izhakian, Tropical arithmetic and matrix algebra, Comm. Algebra 37 (2009) 1445–1468. [4] Z. Izhakian, L. Rowen, Supertropical algebra, Adv. Math. 225 (2010) 2222–2286. [5] Z. Izhakian, L. Rowen, A guide to supertropical algebra, in: Advances in Ring Theory, in: Trends Math., 2010, pp. 283–302.